Question: Simplify; express your answer in exponential form. Assume $q\neq 0, t\neq 0$. $\dfrac{{(q^{-4}t^{4})^{-4}}}{{(q^{2}t^{3})^{-1}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(q^{-4}t^{4})^{-4} = (q^{-4})^{-4}(t^{4})^{-4}}$ On the left, we have ${q^{-4}}$ to the exponent ${-4}$ . Now ${-4 \times -4 = 16}$ , so ${(q^{-4})^{-4} = q^{16}}$ Apply the ideas above to simplify the equation. $\dfrac{{(q^{-4}t^{4})^{-4}}}{{(q^{2}t^{3})^{-1}}} = \dfrac{{q^{16}t^{-16}}}{{q^{-2}t^{-3}}}$ Break up the equation by variable and simplify. $\dfrac{{q^{16}t^{-16}}}{{q^{-2}t^{-3}}} = \dfrac{{q^{16}}}{{q^{-2}}} \cdot \dfrac{{t^{-16}}}{{t^{-3}}} = q^{{16} - {(-2)}} \cdot t^{{-16} - {(-3)}} = q^{18}t^{-13}$